Question

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q is the midpoint of (overline{ru}) and t is the midpoint of (overline{su}). if (rs = w + 27) and (qt = w - 8), what is the value of w?

Explanation:

Step1: Apply the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side. Here, \(QT\) is the mid - segment of \(\triangle RSU\) (since \(Q\) is the midpoint of \(RU\) and \(T\) is the midpoint of \(SU\)), so \(QT=\frac{1}{2}RS\).

Step2: Substitute the given expressions

Given \(RS = w + 27\) and \(QT=w - 8\), substitute into the equation \(QT=\frac{1}{2}RS\). We get \(w - 8=\frac{1}{2}(w + 27)\).

Step3: Solve the equation

Multiply both sides of the equation \(w - 8=\frac{1}{2}(w + 27)\) by \(2\) to eliminate the fraction: \(2(w - 8)=w + 27\).
Expand the left - hand side: \(2w-16 = w + 27\).
Subtract \(w\) from both sides: \(2w-w-16=w - w+ 27\), which simplifies to \(w-16 = 27\).
Add \(16\) to both sides: \(w-16 + 16=27+16\).

Answer:

\(w = 43\)